Related pages

This page describes a slightly different presentation of
affine planes as coordinate
planes over some set K. Let K be a set. We will use
as the set of points of our geometry. Given a set S of
permutations of K, the set of lines is defined to be

We have the following simple theorem.

Theorem. The pair
is an affine plane if and only
if
and the following two conditions hold.

The set S acts sharply 2-transitively on K, i.e. given two pairs
(x1,x2) and (y1,y2) of
distinct points of K there exists exactly one permuation in S which
maps x1 to y1 and x2 to y2,
respectively.

Given an element
of S, and two points x, y in K with
,
there exists exactly one
such that
and